# Time-Frequency Based Channel Estimation for High-Mobility OFDM Systems–Part I: MIMO Case

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## Abstract

Multiple-input multiple-output (MIMO) systems hold the potential to drastically improve the spectral efficiency and link reliability in future wireless communications systems. A particularly promising candidate for next-generation fixed and mobile wireless systems is the combination of MIMO technology with Orthogonal Frequency Division Multiplexing (OFDM). OFDM has become the standard method because of its advantages over single carrier modulation schemes on multipath, frequency selective fading channels. Doppler frequency shifts are expected in fast-moving environments, causing the channel to vary in time, that degrades the performance of OFDM systems. In this paper, we present a time-varying channel modeling and estimation method based on the Discrete Evolutionary Transform to obtain a complete characterization of MIMO-OFDM channels. Performance of the proposed method is evaluated and compared on different levels of channel noise and Doppler frequency shifts.

### Keywords

Orthogonal Frequency Division Multiplex Channel Estimation Doppler Frequency Orthogonal Frequency Division Multiplex System Orthogonal Frequency Division Multiplex Symbol## 1. Introduction

The major challenges in future wireless communications systems are increased spectral efficiency and improved link reliability. The wireless channel constitutes a hostile propagation medium, which suffers from fading (caused by destructive addition of multipath components) and interference from other users. Diversity provides the receiver with several (ideally independent) replicas of the transmitted signal and is therefore a powerful means to combat fading and interference and thereby improve link reliability. Common forms of diversity are space-time diversity [1] and space-frequency diversity [2]. In recent years the use of spatial (or antenna) diversity has become very popular, which is mostly due to the fact that it can be provided without loss in spectral efficiency. Receive diversity, that is, the use of multiple antennas on the receiver side of a wireless link, is a well-studied subject [3]. Driven by mobile wireless applications, where it is difficult to deploy multiple antennas in the handset, the use of multiple antennas on the transmitter side combined with signal processing and coding has become known under the name of space-time coding [4] and is currently an active area of research. The use of multiple antennas at both ends of a wireless link (multiple-input multiple-output (MIMO) technology) has been demonstrated to have the potential of achieving extraordinary data rates [5]. The corresponding technology is known as spatial multiplexing [6] or BLAST [7] and yields an impressive increase in spectral efficiency. Most of the previous work in the area of MIMO wireless has been restricted to narrow-band systems. Besides spatial diversity broadband MIMO channels, however, offer to higher capacity and frequency diversity due to delay spread. Orthogonal frequency division multiplexing (OFDM) significantly reduces receiver complexity in wireless broadband systems. The use of MIMO technology in combination with OFDM, that is, MIMO-OFDM [6], therefore seems to be an attractive solution for future broadband wireless systems [8, 9]. However, intercarrier interference (ICI) due to Doppler shifts, phase offset, local oscillator frequency shifts, and multi-path fading severely degrades the performance of OFDM systems [10]. Most of the channel estimation methods assume a linear time-invariant model for the channel, which is not valid for the next-generation, fast-moving environments [11]. Recently a time-frequency varying MIMO-OFDM channel estimation approach is presented where discrete prolate spheroidal sequences are used to obtain a robust time-varying channel estimator that does not require any channel statistics [12]. A time-varying model of the channel can be obtained by employing time-frequency representation methods. Here we present a time-varying MIMO-OFDM channel estimation based on the discrete evolutionary representation of the channel output. The Discrete Evolutionary Transform (DET) [13] provides a time-frequency representation of the received signal by means of which the spreading function of the multi-path, fading, and frequency selective channel can be modeled and estimated.

The rest of the paper is organized as follows. In Section 2, we give a brief summary of the wireless parametric channel model used in our approach and the MIMO-OFDM communication system. Section 3 presents time-varying modeling and estimation of MIMO-OFDM channels via DET. A time-frequency receiver is also given in Section 3 for the detection of data symbols using estimated channel parameters. In Section 4, we give some simulation results to illustrate the performance of our algorithm for different levels of channel noise and Doppler frequency-shifts and compare with other existing methods. Conclusions are drawn in Section 5.

## 2. MIMO-OFDM System Model

In this section we give a brief introduction to the time-varying, parametric communication channel model used in our work and the MIMO-OFDM signal model.

### 2.1. Parametric Channel Model

where Open image in new window is the speed of light in the transmission medium. In the new generation wireless mobile communication systems, with fast moving objects and high carrier frequencies, Doppler frequency-shifts become significant and have to be taken into consideration. The channel parameters cannot be easily estimated from the impulse response; however the estimation problem can be solved in the time-frequency domain by means of the so called spreading function.

provided that the Doppler frequency shifts are integer multiples of the frequency sampling interval Open image in new window . Open image in new window displays peaks located at the time-frequency positions determined by the delays and the corresponding Doppler frequencies, with Open image in new window as their amplitudes [17]. In our approach, we extract this information from the spreading function of the received signal and then detect the transmitted data symbol.

### 2.2. MIMO-OFDM Signal Model

In an OFDM communication system, the available bandwidth Open image in new window is divided into Open image in new window subchannels. The input data is also divided into Open image in new window -bit parallel bit streams and then mapped onto some transmit symbols Open image in new window drawn from an arbitrary constellation points where Open image in new window is the time index, and Open image in new window , denotes the frequency or subcarrier index.

Some pilot symbols are inserted at some preassigned positions Open image in new window , known to the receiver: Open image in new window Open image in new window where Open image in new window is the number of pilots, and the integer Open image in new window is the distance between adjacent pilots in an OFDM symbol [10]. The Open image in new window OFDM symbol Open image in new window is obtained by taking Open image in new window -point inverse DFT and then adding a cyclic prefix (CP) of length Open image in new window where Open image in new window is chosen such that Open image in new window , and Open image in new window is the time-support of the channel impulse response. This is done to mitigate the effects of intersymbol interference (ISI) caused by the channel dispersion in time.

## 3. Time-Varying Channel Estimationfor MIMO-OFDM Systems

In this section we present a time-frequency procedure to characterize time-varying MIMO-OFDM channels. We also propose a time-frequency receiver that uses the estimated channel fading, delay, and Doppler parameters to recover the transmitted symbols. In the following, we briefly present the Discrete Evolutionary Transform (DET) as a tool for the time-frequency representation of time-varying MIMO-OFDM channels.

### 3.1. The Discrete Evolutionary Transform

where Open image in new window is, in general, a time- and frequency-dependent window.

The DET can be seen as a generalization of the short-time Fourier transform, where the windows are constant. The windows Open image in new window can be obtained from either the Gabor representation that uses nonorthogonal frames or the Malvar wavelet representation that uses orthogonal systems. Details of how the windows can be obtained for the Gabor and Malvar representations are given in [13]. However, for the representation of multipath wireless channel outputs, we consider windows that are adapted to the Doppler frequencies of the channel.

### 3.2. MIMO-OFDM Channel Estimation by Using DET

By comparing Open image in new window and Open image in new window , we observe that the channel parameters Open image in new window , Open image in new window , and Open image in new window calculated from Open image in new window can also be estimated from the downsampled spreading function Open image in new window .

which is the expected result multiplied by Open image in new window . In our estimation procedure, we use windows Open image in new window where Open image in new window is chosen in a discrete set with certain increments, Open image in new window . When Open image in new window coincides with one of the Doppler frequencies in the channel, the spreading function displays a large peak at the time-frequency position Open image in new window , corresponding to delay and Doppler frequency of that transmission path, with magnitude proportional to attenuation Open image in new window . When Open image in new window does not coincide with any of the Doppler frequencies, the spreading function displays a random sequence of peaks spread over all possible delays. Then it is possible to determine a threshold that permits us to obtain the most significant peaks of the spreading function corresponding to possible delays and Doppler frequencies. In our experiments we observed that peaks having amplitudes larger than 65% of the maximum peak are due to an actual transmission; otherwise they are considered as noise. Thus, by searching in the possible Doppler frequency range, we are able to estimate all the parameters of a multi-path, fading, and time-varying MIMO-OFDM channel via the spreading function of the channel.

According to (27), we need the input pilot symbols Open image in new window to estimate the channel frequency response. Here we consider simple, uniform pilot patterns; however improved patterns may be employed as well [11].

### 3.3. Time-Frequency Receiver

where Open image in new window is a Open image in new window matrix; Open image in new window , Open image in new window and Open image in new window are Open image in new window vectors defined by Open image in new window , Open image in new window , and Open image in new window , respectively. Finally, data symbols Open image in new window can be estimated by using a simple time-frequency receiver: Open image in new window .

The exhaustive search for the channel Doppler frequencies may seem to increase the computational cost of the proposed method. However, considering the carrier frequencies and maximum possible velocities in the environment, Doppler frequencies lie in a certain band which can be easily covered by the algorithm. Furthermore, our channel estimation approach does not require any a priori information on the statistics of the channel as in the case of many other channel estimation methods [11].

In the following, we demonstrate time-varying MIMO-OFDM channel estimation performance of our time-frequency-based approach by means of examples.

## 4. Simulations

## 5. Conclusions

In this work, we present a time-varying estimation of MIMO-OFDM channels for high-mobility communication systems by means of discrete evolutionary transform. The main advantage of the proposed method is that it does not assume any statistics on the communication channel. The parametric channel model used in this approach allows us to obtain a two-dimensional representation for the channel and estimate its parameters from the spreading function. We observe that the method is robust against large variations on the channel frequency response, that is, fast fading. Simulations show that our time-frequency-based method has considerably better channel estimation and bit error performance compared to a similar time-frequency varying channel estimation approach [12].

## Notes

### Acknowledgments

This work was partially supported by The Research Fund of The University of Istanbul, Project nos. 6904, 3898, and 4382.

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